Question

In regard to the common belief in a permanent soul, the buddha seems to have taught that

Answer 1

I believe the answer is: There is none

In Buddhism teachings, there is a topic that refers to the doctrine of "non-self" which conveys that there is no unchanging essences in living beings.
Due to this doctrine, Buddhists tend to see that there is no reason to believe the existence of eternal soul that become the basis of their reincarnation belief.